226 problems

Fix an integer $d\geq 1$. For an unordered partition $(d_1,\ldots,d_r)\vdash d$ into strictly positive parts, let $\#\operatorname{Aut}(d_1,\ldots,d_r)$ be the order of its automor…

The big quantum cohomology of G/P is generically semisimple.

Let $X$ be a smooth projective variety with an algebraic action of a complex torus $T_{\mathbb C}$, and let $Y=X/\!/T_{\mathbb C}$ be a smooth GIT quotient with no orbifold singula…

Let $M_N^k$ be the degree-$k$ projective Fano hypersurface considered in the paper, and let $\langle {\cal O}_{h^a}{\cal O}_{h^b}\rangle_0(t^0,t^1,\ldots,t^{N-2})$ denote the pertu…

Let $M$ be a Calabi–Yau threefold, and let $\beta$ be a nonzero curve class in $M$. The symbols $\operatorname{Z_{GW}^{\prime}}(M;u)_\beta$ and $\operatorname{Z_{PT}}(M;q)_\beta$ d…

Let $\mathcal{X}$ be an orbifold with projective coarse moduli space $X$, and let $\pi:Y\to X$ be a crepant resolution. Suppose that the big quantum products $\bullet_\tau$ for…

Let $X_k=\mathcal O(k)\oplus\mathcal O(-2-k)\rightarrow\mathbb{P}^1$ carry the diagonal torus action $\lambda_1=\lambda_2$, and let…

Let $X_k=\mathcal O(k)\oplus\mathcal O(-2-k)\rightarrow\mathbb{P}^1$ carry the antidiagonal torus action $(\lambda,-\lambda)$ on the bundle, and let $t_k$ denote the Kähler class o…

Let $\sum_i c_i\Gamma_i=0$ be a tautological equation in codimension $k$ on $\overline{M}_{g,n}$, and let $\mathfrak{r}_l$ be the operators sending decorated graphs on…

Let $\pi\colon\mathcal W\to\mathbb P^{1}$ be a family of quintic mirror threefolds. Let $N_g(d)$ be the genus-$g$ Gromov–Witten invariant of degree $d$ of a general quintic threefo…

Let $Y$ be a variety with transversal $A_n$-singularities, $n\geq 2$, and let $\rho:Z\to Y$ be its crepant resolution. Let $K$ be the line bundle on the singular locus appearing in…

For a Calabi–Yau $A_\infty$ category, let the Gromov–Witten potential define a left ideal in the sheaf of Weyl algebras associated to its periodic cyclic chain complex, and let thi…

Sine formula conjecture. For all $d\geq 1$,

Golyshev's conjecture. The family of counting equations for $X$ contains an equation

Let $X$ be a Calabi–Yau threefold, meaning that $K_X$ is trivial. For curve classes $\beta\ne0$, let $Z^{\mathrm{DT}}(X)'$ be the reduced Donaldson–Thomas partition function, and l…

J-function conjecture. $J^{X//G}(t_0,\mathbf t,\hbar)$ is obtained from $I$ by an explicit change of variables (the “mirror transformation”). If $X//G$ is Fano of index at least…

Let $P_g$ be the set of partitions of $g$, let $S(\sigma_g)$ be the partition coefficient defined by the generating function in the paper, let $\tilde{L}^{N,k,d_i}_{1+(k-N)d_i}$ be…

Let $\sigma_m=d_1+\cdots+d_{l(\sigma_m)}$ be a partition, let $V_{d-m}^{N,k,d}(n;\sigma_m)$ be the normalized virtual Gromov–Witten invariant defined in the paper, and let…

For $g\geq 0$ and $n\geq 1$, let $Z_{g;n}(q):=\sum_dN_{g,d;n}q^d$, where $N_{g,d;n}:=\sum_{(\beta,\sigma)=d,\,(\beta,F)=n}N_g(\beta)$. Let $E_2(q),E_4(q),E_6(q)$ be Eisenstein seri…

Hurwitz-number congruence conjecture. If $d$ is not divisible by $4$, $6$, or $9$, then

Let $x$ and $y$ be the algebraic coordinates used for the two-parameter model, and let $dis_0$ denote the discriminant component defined by … For $g\geq 2$, define the propagators…

Givental's conjecture. The generating function $G(\{T_k^+\},\{0\})$ is equal to one, or equivalently

Let $\mathcal{Z}$ be the string partition function, let $x$ be the string coupling parameter, and let $F_0^{(0)}$ and $F_1^{(0)}$ be the classical genus-zero and genus-one potentia…

The chamber formula conjecture. In the fundamental chamber,

Let $S$ be a symplectic surface, and let $\beta\in H_2(S,\mathbb{Z})$ be an effective curve class. For every positive integer $k$ dividing $\beta$, let … be a degree-preserving…